A few riddles behind Rolle's theorem

2003/02/18 by B. Shapiro, Boris Shapiro, Shapiro, B. +3 · 1 citation
Mathematics · #26A06 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematics and Applications #math.CA #msc:26A06

paper · pdf · doi:10.48550/arxiv.math/0302215

7 pages, 3 figures, easy reading for undergraduates, Title changed, some typos fixed

openalex publication_date 2003/02/18 · arxiv created 2005/12/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We call a smooth function of one variable a degree n pseudopolynomial if its n-th derivative has no (real) zeros. An n pseudopolynomial is called hyperbolic if it has exactly n simple zeros. In this short note we describe the necessary and sufficient conditions on the arrangements of 6 points consisting of 3 zeros of pseudopolynomials of degree 3, two zeros of their 1st derivatives and 1 zero of their second derivatives. Besides the standard Rolle's inequalities the restrictions include additional quadratic inequalities of geometric origin. This text is an easy reading accessible for undergraduates. We formulated also two questions not solved yet.

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• Midpoints and critical points

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